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A003987 Table of n XOR m (or Nim-sum of n and m) read by antidiagonals, i.e., with entries in the order (n,m) = (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ... 183

%I

%S 0,1,1,2,0,2,3,3,3,3,4,2,0,2,4,5,5,1,1,5,5,6,4,6,0,6,4,6,7,7,7,7,7,7,

%T 7,7,8,6,4,6,0,6,4,6,8,9,9,5,5,1,1,5,5,9,9,10,8,10,4,2,0,2,4,10,8,10,

%U 11,11,11,11,3,3,3,3,11,11,11,11,12,10,8,10,12,2,0,2,12,10,8,10,12,13,13,9,9

%N Table of n XOR m (or Nim-sum of n and m) read by antidiagonals, i.e., with entries in the order (n,m) = (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...

%C Another way to construct the array: construct an infinite square matrix starting in the top left corner using the rule that each entry is the smallest nonnegative number that is not in the row to your left or in the column above you.

%C After a few moves the [symmetric] matrix looks like this:

%C 0 1 2 3 4 5 ...

%C 1 0 3 2 5 ...

%C 2 3 0 1 ?

%C 3 2 1

%C 4 5 ?

%C 5

%C The ? is then replaced with a 6.

%D E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.

%D J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

%D Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, date?

%D D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998; see p. 190. [From _N. J. A. Sloane_, Jul 14 2009]

%D R. K. Guy, Impartial games, pp. 35-55 of Combinatorial Games, ed. R. K. Guy, Proc. Sympos. Appl. Math., 43, Amer. Math. Soc., 1991.

%H T. D. Noe, <a href="/A003987/b003987.txt">Rows n = 0..100 of triangle, flattened</a>

%H J.-P. Allouche and J. Shallit, <a href="https://doi.org/10.1016/S0304-3975(03)00090-2">The Ring of k-regular Sequences, II</a>, Theoret. Computer Sci., 307 (2003), 3-29.

%H Rémy Sigrist, <a href="/A003987/a003987.png">Colored representation of T(x,y) for x = 0..1023 and y = 0..1023</a> (where the hue is function of T(x,y) and black pixels correspond to zeros)

%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).

%H <a href="/index/Ni#Nimsums">Index entries for sequences related to Nim-sums</a>

%F T(2i,2j) = 2T(i,j), T(2i+1,2j) = 2T(i,j) + 1.

%e Table begins

%e 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...

%e 1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, ...

%e 2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, ...

%e 3, 2, 1, 0, 7, 6, 5, 4, 11, 10, ...

%e 4, 5, 6, 7, 0, 1, 2, 3, 12, ...

%e 5, 4, 7, 6, 1, 0, 3, 2, ...

%e 6, 7, 4, 5, 2, 3, 0, ...

%e 7, 6, 5, 4, 3, 2, ...

%e 8, 9, 10, 11, 12, ...

%e 9, 8, 11, 10, ...

%e 10, 11, 8, ...

%e 11, 10, ...

%e 12, ...

%e ...

%e The first few antidiagonals are

%e 0;

%e 1, 1;

%e 2, 0, 2;

%e 3, 3, 3, 3;

%e 4, 2, 0, 2, 4;

%e 5, 5, 1, 1, 5, 5;

%e 6, 4, 6, 0, 6, 4, 6;

%e 7, 7, 7, 7, 7, 7, 7, 7;

%e 8, 6, 4, 6, 0, 6, 4, 6, 8;

%e 9, 9, 5, 5, 1, 1, 5, 5, 9, 9;

%e 10, 8, 10, 4, 2, 0, 2, 4, 10, 8, 10;

%e 11, 11, 11, 11, 3, 3, 3, 3, 11, 11, 11, 11;

%e 12, 10, 8, 10, 12, 2, 0, 2, 12, 10, 8, 10, 12;

%e ...

%e [Symmetric] matrix in base 2:

%e 0 1 10 11 100 101, 110 111 1000 1001 1010 1011 ...

%e 1 0 11 10 101 100, 111 110 1001 1000 1011 ...

%e 10 11 0 1 110 111, 100 101 1010 1011 ...

%e 11 10 1 0 111 110, 101 100 1011 ...

%e 100 101 110 111 0 1 10 11 ...

%e 101 100 111 110 1 0 11 ...

%e 110 111 100 101 10 11 ...

%e 111 110 101 100 11 ...

%e 1000 1001 1010 1011 ...

%e 1001 1000 1011 ...

%e 1010 1011 ...

%e 1011 ...

%e ...

%p nimsum := proc(a,b) local t1,t2,t3,t4,l; t1 := convert(a+2^20,base,2); t2 := convert(b+2^20,base,2); t3 := evalm(t1+t2); map(x->x mod 2, t3); t4 := convert(evalm(%),list); l := convert(t4,base,2,10); sum(l[k]*10^(k-1), k=1..nops(l)); end; # memo: adjust 2^20 to be much bigger than a and b

%p AT := array(0..N,0..N); for a from 0 to N do for b from a to N do AT[a,b] := nimsum(a,b); AT[b,a] := AT[a,b]; od: od:

%p # alternative:

%p read("transforms") :

%p A003987 := proc(n,m)

%p XORnos(n,m) ;

%p end proc: # _R. J. Mathar_, Apr 17 2013

%p seq(seq(Bits:-Xor(k,m-k),k=0..m),m=0..20); # _Robert Israel_, Dec 31 2015

%t Flatten[Table[BitXor[b, a - b], {a, 0, 10}, {b, 0, a}]] (* BitXor and Nim Sum are equivalent *)

%o (PARI) tabl(nn) = {for(n=0, nn, for(k=0, n, print1(bitxor(k, n - k),", ");); print(););};

%o tabl(13) \\ _Indranil Ghosh_, Mar 31 2017

%o (Python)

%o for n in range(14):

%o print([k^(n - k) for k in range(n + 1)]) # _Indranil Ghosh_, Mar 31 2017

%Y Initial rows are A001477, A004442, A004443, A004444, etc. Cf. A051775, A051776.

%Y Cf. A003986 (OR), A004198 (AND), A221146 (carries).

%Y Antidiagonal sums are in A006582.

%K tabl,nonn,nice,look

%O 0,4

%A _Marc LeBrun_

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Last modified July 25 23:39 EDT 2021. Contains 346294 sequences. (Running on oeis4.)